Computing and Visualization in Science

, Volume 16, Issue 2, pp 59–76 | Cite as

Adaptive AMG with coarsening based on compatible weighted matching

  • Pasqua D’Ambra
  • Panayot S. Vassilevski


We introduce a new composite adaptive Algebraic Multigrid (composite \(\alpha \)AMG) method to solve systems of linear equations without a-priori knowledge or assumption on characteristics of near-null components of the AMG preconditioned problem referred to as algebraic smoothness. Our version of \(\alpha \)AMG is a composite solver built through a bootstrap strategy aimed to obtain a desired convergence rate. The coarsening process employed to build each new solver component relies on a pairwise aggregation scheme based on weighted matching in a graph, successfully exploited for reordering algorithms in sparse direct methods to enhance diagonal dominance, and compatible relaxation. The proposed compatible matching process replaces the commonly used characterization of strength of connection in both the coarse space selection and in the interpolation scheme. The goal is to design a method leading to scalable AMG for a wide class of problems that go beyond the standard elliptic Partial Differential Equations (PDEs). In the present work, we introduce the method and demonstrate its potential when applied to symmetric positive definite linear systems arising from finite element discretization of highly anisotropic elliptic PDEs on structured and unstructured meshes. We also report on some preliminary tests for 2D and 3D elasticity problems as well as on problems from the University of Florida Sparse Matrix Collection.


Adaptive AMG Weighted matching Strength of connection Compatible relaxation 

Mathematics Subject Classification

65F10 65N55 



We wish to thank Bora Uçar for stimulating discussions and his advice on available algorithms and software for computing weighted matching in bipartite and general graphs. We also thank one of the authors of MFEM, Tzanio Kolev for his kind help in installing and using MFEM. Finally, we thank one of the reviewers for motivating us to run more examples and use one more version of coarsening that demonstrated better the potential of the proposed composite \(\alpha \)AMG solver.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute for High-Performance Computing and NetworkingNational Research Council of ItalyNaplesItaly
  2. 2.Center for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermoreUSA

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